Machine learning and optimization-based approaches to duality in statistical physics

TL;DR

This paper introduces a machine learning framework that uses neural networks and optimization to discover dualities in statistical physics models, successfully rediscovering known dualities and exploring new variants.

Contribution

It presents a novel neural network-based approach to automatically identify dualities and dual Hamiltonians in lattice models, advancing the methodology for duality discovery.

Findings

Successfully rediscovered Kramers-Wannier duality for 2d Ising model

Demonstrated the framework's ability to find dualities in deformed models

Proposed alternative methods leveraging topological features

Abstract

The notion of duality -- that a given physical system can have two different mathematical descriptions -- is a key idea in modern theoretical physics. Establishing a duality in lattice statistical mechanics models requires the construction of a dual Hamiltonian and a map from the original to the dual observables. By using simple neural networks to parameterize these maps and introducing a loss function that penalises the difference between correlation functions in original and dual models, we formulate the process of duality discovery as an optimization problem. We numerically solve this problem and show that our framework can rediscover the celebrated Kramers-Wannier duality for the 2d Ising model, reconstructing the known mapping of temperatures. We also discuss an alternative approach which uses known features of the mapping of topological lines to reduce the problem to optimizing the couplings in a dual Hamiltonian, and explore next-to-nearest neighbour deformations of the 2d Ising duality. We discuss future directions and prospects for discovering new dualities within this framework.